Simplify (2^-2)^(3/2)

The given problem is a mathematical expression that involves exponents and requires simplification. Specifically, it asks you to apply the rules of exponents to simplify the expression (2^-2)^(3/2), which is a power raised to another power. This involves manipulating the base and the exponent according to the laws of exponents to rewrite the expression in a simpler or more standard form.

${((2^{- 2}))}^{\frac{3}{2}}$

Answer

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Solution:

Step 1: Exponent Multiplication

Combine the exponents in ${(2^{- 2})}^{\frac{3}{2}}$ by multiplying them together.

Step 1.1: Apply the Power to a Power Rule

Utilize the rule ${(a^{m})}^{n} = a^{m n}$ to simplify the expression to $2^{- 2 \cdot \frac{3}{2}}$ .

Step 1.2: Simplify the Exponent

Identify and simplify common factors in the exponent.

Step 1.2.1: Extract the Factor of 2

Extract the factor of 2 from the exponent to get $2^{2 \cdot (- 1) \cdot \frac{3}{2}}$ .

Step 1.2.2: Simplify the Common Factors

Simplify the common factors to get $2^{2 \cdot - 1 \cdot \frac{3}{2}}$ .

Step 1.2.3: Finalize the Exponent

Finalize the simplification of the expression to $2^{- 1 \cdot 3}$ .

Step 1.3: Complete the Multiplication

Complete the multiplication of the exponents to get $2^{- 3}$ .

Step 2: Apply the Negative Exponent Rule

Convert the negative exponent to a fraction using the rule $b^{- n} = \frac{1}{b^{n}}$ to obtain $\frac{1}{2^{3}}$ .

Step 3: Calculate the Power of 2

Evaluate the power of 2 to get $\frac{1}{8}$ .

Step 4: Present the Final Result

The simplified expression can be presented in various forms.

Exact Form: $\frac{1}{8}$ Decimal Form: $0.125$

Knowledge Notes:

To solve the given problem, several mathematical rules and properties are applied:

Power to a Power Rule: When you raise a power to another power, you multiply the exponents. The general form of this rule is $(a^{m})^{n} = a^{m n}$ .
Simplifying Exponents: When simplifying expressions with exponents, look for common factors that can be canceled out to simplify the calculation.
Negative Exponent Rule: A negative exponent indicates that the base is on the wrong side of a fraction line. To make the exponent positive, you can take the reciprocal of the base and then raise it to the absolute value of the exponent. The rule is $b^{- n} = \frac{1}{b^{n}}$ .
Evaluating Powers: To evaluate a power such as $2^{3}$ , you multiply the base (2) by itself as many times as the exponent indicates (3 times in this case), which equals $2 \cdot 2 \cdot 2 = 8$ .
Exact vs Decimal Form: The exact form of a number is the form that represents the value without any approximation, while the decimal form is a numerical representation that may be rounded or truncated.

In this problem, we applied the power to a power rule to combine the exponents, simplified the resulting expression, used the negative exponent rule to rewrite the expression as a fraction, and finally evaluated the power of 2 to find the exact and decimal form of the result.